MATHEMATICS RESEARCH PROJECT IDEAS
SUITABLE FOR HIGH SCHOOL AND COLLEGE STUDENTS

This list is a copy of the list Possible Science Fair Mathematics Projects
which is maintained by
Afton H. Cayford ,
at The University of British Columbia.

What follows is a selection of ideas for science projects. In most cases only
a very brief outline is
presented (sometimes with a reference) in order to leave students plenty of
scope for
what they do. It is
not expected that the problem stated will necessarily be the project. These
are
ideas intended
to get people thinking (they are in no particular order).

1.
At certain times charities call
households offering to pick-up used items for sale in their stores.
They often do a particular geographical area at a time. Their problem, once
they know where the pick-ups are, is to decide on the most efficient routes to
make the collection. Find out how they do this and
investigate improving their procedure. A similar question can be asked
about snow plows clearing city streets, or garbage collection.
References: Euclidean tours, chinese postman problem - information can
be
found in
most books on graph theory but one of particular interest is ``Introduction to
Graph Theory'' by G. Chartrand.

2.
How should one to locate ambulance stations, so as to best serve
the needs of the community? The reference given above may help.

3.
An International Food Group consists of
twenty couples who meet four times a year for a meal. On each occasion, four
couples meet at each of five houses. The members of the group get along very
well together; nonetheless, there is always a bit of discontent during the year
when some couples meet more than once! Is it possible to plan four evenings
such that no two couples meet more than
once? There are many problems like this. They are called combinatorial
designs. Investigate others.

4.
How does the NBA work out the basketball schedule? How would you do such
a
schedule bearing in mind distances between locations of games, home team
advantage
etc.? Could you devise a good schedule for one of your local competitions?

5.
How do major hospitals schedule the use of operating theatres? Are they
doing
it the best way possible so that the maximum number of operations are
done each day?

6.
Investigate ``big'' numbers. What is a big number? The following
examples might guide your investigation. A bank is robbed of 1 million
loonies. How long it would take to move that many? How much it would
weigh? How much space would it take up? How big a swimming pool do you need to
contain
all the blood in the world? Is 10100 very big? What is the biggest
number anyone has ever written down (check the Guiness book of world records
over the last few years)? How did this number come about?

7.
Build a phsical model based on dissections to prove the Pythagorean
Theorem.
Build an exhibit on the Pythagorean theorem but with "The semicircle on the
hypotenuse ..."

8.
What is the fewest number of colours needed to colour any map if the
rule is that
no two countries with a common border can have the same colour. Who discovered
this?
Why is the proof interesting? What if Mars
is also divided into areas so that these areas are owned by different
countries on earth. They too are coloured by the same rule but the
areas there must be coloured by the colour of the country they belong to.
How many colours are now needed?
Reference: Joan Hutchinson, ...

9.
Study the golden mean, its appearance in art, architecture, biology,
and
geometry, and it's connection with continued fractions, fibonacci numbers.
What
else can you find out? What is the Golden Mean?

11.
Study the cycloid curve: its tautochrone and brachistochrone properties
and its history. Build models.

12.
Infinity comes in different ``sizes''. What does this mean? How can it
be
explained?
References: Refer to either of the Dover paperbacks, ``Theory of Sets''
by Kamke, and ``The
Continuum and other types of Serial Order'' by Huntington, or any book on
Set Theory.

13.
Investigate visual representations of different finite
numbers. For example, if p is a prime with 100 digits, then if
1 and p are on the same line segment, with p say 6 inches to the
right of 1, then p1/2, the square root of p,
is about 10-50 inches to the
right of 1, less than one atom away. (And it's by inspecting the
lattice points in the p1/2 x p1/2
array that one proves
that p is the sum of two squares!) Investigate further.

15.
Study games and winning strategies - maybe explore a game where the
winning
strategy is not known. Analyze subtraction games (nim-like games in which the
two players
alternately take a number of beans from a heap, the numbers being restricted
to a given subtraction set).
References: E.R. Berlekamp, J.H. Conway, R.K. Guy, ``Winning Ways'',
Academic
Press, London (this book contains hundreds of othr games for which the complete
analysis is unknown eg. Toads and Frogs) ; R. Guy (editor), ``Combinatorial
Games'', Proceeedings of Symposia in Applied Math,
AMS publication (pay special attention to the last section where lots of
questions are
asked).

16.
Most computers these days can handle sound one way or another. They
store
the sound as a sequence of numbers. Lots of numbers. 40,000 per second,
say. What happens when you play around with those numbers? eg. Add 10 to each
number. Multiply each number by 10. Divide by 10. Take absolute values.
Take one sound, and add it to another sound (i.e. add up corresponding
pairs of numbers in the sequences). Multiply them. Divide them. Take one
sound, and add it to shifted copies of itself. Shuffle the numbers in the
sequence. Turn them around backwards. Throw out every third number. Take
the sine of the numbers. Square them.
For each mathematical operation, you can play the resulting sound on the
computers speakers, and hear what change has occurred. A little bit of
programming, and you can get some very bizarre effects.
Then try to make sense of this from some sort of theory of signal
processing.

20.
Make a family of polyhedra, e.g., the Archimedean solids, or Deltahedra
(whose faces are all equilateral triangles), or equilateral zonohedra, or, for
the very ambitious, the 59 Isocahedra.
Reference: See any Coxeter revision of Rouse
Ball's ``Mathematical Recreations and Essays'' (which is full of many ideas).
There's also Coxeter, DuVal, Flather and Petrie, ``The 59 Icosahedra'', U of
Toronto
Press; Magnus J Wenninger, ``Polyhedron Models'', Cambridge, 1971; and Doris
Schattschneider and Wallace Walker, ``M.C. Escher Kaleidocycles'', Pomegranate
Art Books, 1987.

21.
Find as many triangles as you can with integer sides and a simple linear
relation between the angles. What about the special case when the triangle is
right-angled?

22.
Find out all you can about the Fibonacci Numbers, 0, 1, 1, 2, 3, 5, 8,
...

23.
Find out all you can about the Catalan Numbers, 1, 1, 2, 5, 14, 42, ...

24.
What is Morley's triangle? Draw a picture of the 18 Morley triangles
associated with a given triangle ABC. Find
the 18 more for each of the triangles BHC, CHA, AHB, where H is the
orthocentre of ABC. Discover the relation with the 9-point circle and deltoid
(envelope of
the Simson or Wallace line).

25.
What is a hexaflexagon? Make as may different ones as you can.
What is going on?
Reference: Martin Gardner, ``Hexaflexagons and other Mathematical
Diversions'',
Univ. of Chicago Press, 1988.

26.
Investigate trianglar numbers. If that's not enough, do squares,
pentagonal numbers, hexagonal numbers, etc Venture into the third
and even the fourth dimension. Reference: Conway and Guy, ``The Book of
Numbers'', Springer, Copernicus Series, 1996, Chapter 2.

27.
Ten frogs sit on a log - 5 green frogs on one side and 5 brown frogs on
the
other with an empty seat separating them. They decide to switch places. The
only moves permitted are to jump over one frog of a different colour into an
empty space or to jump into an adjacent space. What is the minimum number of
moves? What if there were 100 frogs on each side?
Coming up with the answers reveals interesting patterns depending on whether
you focus on colour of frog, type of move, or empty space. Proving it works
is interesting also - it can lead to recursion, there is also a simple proof
that is not immediately obvious when you start.
Look for and explore other questions like this - one of the most famous is the
Tower of Hanoi.

28.
Investigate the creation of secret codes (ciphers). Find out where
they
are used (today!) and how they are used. Look at their history. Build your
own
using prime numbers.
Reference: M. Fellows and N. Koblitz.
``Kid krypto.'' Proc. CRYPTO '92, Springer-Verlag, Lecture Notes in Computer
Science
vol. 740 (1993), 371-389.

29.
There is a well-known device for illustrating the binomial
distribution. Marbles are dropped through the top and encounter a number
of pins before dropping into cells where they are distributed according to
the binomial distribution. By changing the position of the pins one
should be able to get other kinds of distributions (bimodal, skewed,
approximately rectangular, etc.) Explore.

30.
Build rigid and nonrigid geometric structures. Explore them. Where are
rigid structures used? Find unusual applications.
This could include an illustration of the fact that the midpoints of the
sides of a quadrilateral form a parallelogram (even when the quadrilateral
is not planar). Are there similar things in three dimensions?

31.
Build a true scale model of the solar system
- but be careful because it cannot be
contained within the confines of an exhibit. Illustrate how you would
locate it in your town. Maybe even do so!!

32.
Build models to illustrate asymptotic results such as Stirling's
formula
or the prime number theorem.

33.
What is/are Napier's bones and what can you do with it/them?

34.
Covering a chessboard with dominoes so that no two dominoes overlap and
no square on the chessboard is uncovered. Consider (a) a full chessboard, (b)
a chessboard with one square removed (impossible - why?),
(c) a chessboard with
two adjacent corners removed, (d) one with two opposite corners removed
(possible or impossible?),
(e) A chessboard with any two squares removed. What about using shapes other
than dominoes (eg 3 one-by-one
squares joined together)? What about chessboards of different dimensions?
Reference: ``Polyominoes'' by Solomon W. Golumb, pub. Charles Scribner's
Sons

35.
Build models showing that parallelograms with the same base
and height have the same areas (is there a 3-dimensional analogue?).
This can lead to a purely visual proof of the Pythagorean theorem.
The formula for the area of a circle can also be
presented in this way.
R eference: H. R. Jacobs,`` Mathematics a Human
Endeavor'', 3rd ed, p 38)

36.
Use Monte Carlo methods to find
areas (rather than using random numbers, throw
a bunch of small objects onto the required area and count the numbers of
objects inside the area as a fraction of the total in the rectangular
frame) or to estimate pi.

37.
Find pictures which show that
1 + 2 + ... + n = (1/2)n(n+1),
that
12 + 22 + ... + n2 = (1/6)n(n+1)(2n+1)
and that
13 + 23 + ... + n3 =
(1 + 2 + ... + n)2.
How many other ways
can you find to prove these identities? Is any one of them ``best''?

38.
What is fractal dimension? Investigate it by exaomining examples showing
what happens
when you double the scale to (a) lines (b) areas (c) solids (d) the Koch curve.

39.
Knots. What happens when you put a knot in a
strip of paper and flatten it carefully? When is what appears to be a knot
really a
knot? Look at methods for drawing knots.

40.
Is there an algorithm for getting out of 2-dimensional mazes? What about
3-dimensional? Look at the history of mazes (some are extraordinary). How
would you
go about finding someone who is lost in a maze (2 or 3 dimensional) and
wandering
randomly? How many people would you need to find them?

41.
Investigate the history of pi and the many ways in which it can be
approximated.
Calculate new digits of Pi - see Peter Borwein's
homepage
to discover what this means.

42.
What is game theory all about and where is it applied?

43.
Construct a Kaleidoscope. Investigate its history and the mathematics
of
symmetry.

44.
Consider tiling the plane using shapes of the same size. What's possible
and what isn't.
In particular it can be shown that any 4-sided shape can tile the plane.
What about 5 sides? Look for books and articles by Grunbaum and Shepherd, and
check the Martin Gardner books.

45.
Explore Penrose tiles and discover why they are of interest.

46.
Investigate the Steiner problem - one application of which is concerned
with
the location of telphone exchanges to minimize costs.

47.
Look for new strategies for solving the travelling salesman problem.

50.
The Art Gallery problem: What is the least number of guards required to
watch
over all paintings in an art gallery? The guards are positioned at specific
locations and collectively must have a direct line of sight to every
point on the walls.
Reference: Alan Tucker, ``The Art Gallery Problem'', Math Horizons,
Spring,
1994, p24-26

51.
The Parabolic Reflector Microphone is used at sporting events when
you want to be able to
hear one person in a noisy area. Investigate this; explaining the mathematics
behind what is happening.

52.
There is a traditional Chinese way of illustrating the Pythagorean
theorem
using paper. Investigate and make models.

53.
Use PID (proportional-integral-differential) controllers and
oscilloscopes
to demonstrate the integration and differentiation of
different functions.

54.
Try the "Monty Hall" effect. Behind one of three doors there is a prize.
You pick
door #1, he shows you that the prize wasn't behind door #2 and
then gives you the choice of switching to door #3 or staying with #1,
what should you do? Why should you switch? Make an exhibit and run trials to
``show'' this is so. Find the mathematical reason for the switch.

55.
Look at the ways different bases are used in our
culture and how they have been used in other cultures. Collect
examples: time, date etc. Demonstrate how
to add using the Mayan base 20, maybe compare to trying to
add with Roman numerals (is it even possible?)

56.
Explore the history and use of the Abacus.

57.
Investigate card tricks. Some of the best in the world were designed
by
the mathematician/statistician Persi Diaconis.
Reference: Don Albers, `` Professor of (Magic) Mathematics'', Math
Horizons, February 1995, p11-15

58.
Explore magic tricks based in Mathematics (again see the article about
Persi Diaconis).

59.
Investigate compass and straight-edge constructions - showing what's
possible and discussing what's not. For example, given a line segment of
length one can you use them
straight edge and compass to ``construct'' all the radicals?